On Poincar\'e lemma or Volterra theorem about differential forms and cohomology groups

TL;DR

This paper provides direct proofs of the Poincaré lemma for differential forms, explores its consequences, and discusses connections with various cohomology theories, emphasizing its fundamental role in differential geometry.

Contribution

It offers new direct proofs of the Poincaré lemma and investigates its implications and links with multiple cohomology frameworks.

Findings

Direct proofs of the Poincaré lemma presented.

Connections with Cech-De Rham-Dolbeault cohomologies analyzed.

Implications for local exactness of closed forms discussed.

Abstract

The Poincar\'{e} lemma (or Volterra theorem) is of utmost importance both in theory and in practice. It tells us every differential form which is closed, is locally exact. In other words, on a contractible manifold all closed forms are exact. The aim of this paper is to present some direct proofs of this lemma and explore some of its numerous consequences. Some connections with Cech-De Rham-Dolbeault cohomologies, $\overline{\partial}$-Poincar\'{e} lemma or Dolbeault-Grothendieck lemma are given.